Isolation critical graphs under multiple edge subdivision

This paper introduces the notion of an $(ι,q)$-critical graph. The isolation number of a graph $G$, denoted by $ι(G)$ and also known as the vertex-edge domination number of $G$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the subgraph induced by the set of vertices that are not in the closed neighbourhood of $D$ has no edges. A graph $G$ is $(ι,q)$-critical if every subdivision of $q$ edges of $G$ gives a graph whose isolation number is greater than $ι(G)$, and $G$ has $q-1$ edges such that subdividing them gives a graph whose isolation number is $ι(G)$. We show that an $(ι,q)$-critical graph exists for every integer $q \ge 1$. We prove that if $G$ is a connected $m$-edge non-star graph, then $G$ is $(ι,q)$-critical for some $q \le m - 1$. We show that this bound is best possible. We provide a general characterization of $(ι,1)$-critical graphs as well as a constructive characterization of $(ι,1)$-critical trees, demonstrating that $(ι,1)$-criticality can be checked in linear time for trees.